What is the difference between logarithmic decay vs exponential decay? I



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What is the difference between logarithmic decay vs exponential decay?
I am a little unclear on whether they are distinctly different or whether this is a 'square is a rectangle, but rectangle is not necessarily a square' type of relationship.

Answer & Explanation



Beginner2022-07-10Added 16 answers

The "Square is a rectangle" relationship is an example where the square is a special case of a rectangle.
"Exponential decay" gets its name because the functions used to model it are of the form f ( x ) = A e k x + C where A > 0 and k < 0. (Other k's above 0 yield an increasing function, not a decaying one.)
Similarly for "logarithmic decay," it gets its name since its modeled with functions of the form g ( x ) = A ln ( x ) + C where A < 0
These two families of functions do not overlap, so neither is a special case of the other. The giveaway is that the functions with ln ( x ) aren't even defined on half the real line, whereas the exponential ones are defined everywhere.


Beginner2022-07-11Added 6 answers

The natural logarithm and exponential are inverses of one another, so the associated slopes will also be inverses. If you put exponentially decaying data on a log plot, i.e. log of the exponential decaying data with the same input, you get a linear plot. If you put the logarithmic decaying plot on an exponential plot (exponential of the data), you get a linear plot, so the way they are decaying is exactly opposite.

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